How do I stop my text messages from appearing on my Macbook?

Are you getting iMessages or SMS messages on your MacBook Air?

For SMS messages, you need to disable the Text Message Forwarding option on your iPhone. That is, tap on Settings > Messages > Text Message Forwarding, and move the slider to the left.

For iMessages, run the Messages app, then from the Messages' menu bar, click on Messages > Preferences. Click on the Accounts tab to select it, and then, disable any messaging accounts that you see there.

Under your Apple ID settings on your iPhone I believe you can link and unlink devices from your Apple ID. Hope this helps.